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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Fall 2010 • Question 1

Question

Math 6010 • Linear Models Fall 2010 • Question 1

Let \(X_1,X_2,\ldots \) be independent identically distributed random variables with characteristic function \(\varphi \). Let \(N\) be a random variable with distribution \[ \mathbb {P}\{N=k\}=\frac {1}{2^k},\qquad k=1,2,\ldots . \] It is assumed that \(\{X_i,\ i\geq 1\}\) and \(N\) are independent.

(a) Compute the characteristic function of \(Y=X_1+\cdots X_N\).

(b) Can you weaken the condition that \(\{X_i,\ i\geq 1\}\) and \(N\) are independent so that the formula obtained in part (a) remains true?

Raw LaTeX

Let \(X_1,X_2,\ldots\) be independent identically distributed random
variables with characteristic function \(\varphi\). Let \(N\) be a random variable
with distribution
\[
  \Prob\{N=k\}=\frac{1}{2^k},\qquad k=1,2,\ldots.
\]
It is assumed that \(\{X_i,\ i\geq1\}\) and \(N\) are independent.

\textbf{(a)} Compute the characteristic function of \(Y=X_1+\cdots X_N\).

\textbf{(b)} Can you weaken the condition that \(\{X_i,\ i\geq1\}\) and \(N\) are
independent so that the formula obtained in part (a) remains true?

Topic Tags

Characteristic Functions and Fourier Inversion Covariance and Independence Probability Inequalities and Tail Bounds Uniform Integrability and Truncation Methods